No Need for an Oracle: The Nonparametric Maximum Likelihood Decision in the Compound Decision Problem Is Minimax

We discuss the asymptotics of the nonparametric maximum likelihood estimator (NPMLE) in the normal mixture model. We then prove the convergence rate of the NPMLE decision in the empirical Bayes problem with normal observations. We point to (and heavily use) the connection between the NPMLE decision and Stein unbiased risk estimator (SURE). Next, we prove that the same solution is optimal in the compound decision problem where the unobserved parameters are not assumed to be random. Similar results are usually claimed using an oracle-based argument. However, we contend that the standard oracle argument is not valid. It was only partially proved that it can be fixed, and the existing proofs of these partial results are tedious. Our approach, on the other hand, is straightforward and short.